Jane asked John if he had said anything about her to another employee. He said he did not. He asked Jane who told him that. Jane refuses to tell him who. Jane stated that his word is good enough for her. John sends me (HR) a written statement and demands an investigation on why employees are gossiping about him.
Oh blech. Here we are in 7th grade all over again. A manager wants an investigation about why employees are gossiping about him? Seriously. Tell him that it would be worth investigating if employees weren't gossiping about him.
And I'm concerned about my spelling of gossiping. It looks funny and I think it should have two ps. But it doesn't; it only has one. Weird.
For the record, I'm on Jane's side here. Instead of engaging in the gossip herself, she went to the supposed source and asked for clarification. She is also trying to end the drama right now by not tattling on whoever said whatever. I usually advise people to ask people directly if they want to know what someone thinks. Good for Jane.
Now, as your John. I have a strong suspicion that he lied to Jane. I bet he has been talking about her and now he's all freaked out because one of the people he "trusted" has now blabbed, but he's blabbed to so many people that he has no idea who to yell at. (No idea at whom to yell? It's grammar day here at EHRL!)
But, even if he didn't, his reaction is over the top. He doesn't know his people. He doesn't trust his people. He's blown up what could easily have been a misunderstanding to a full fledged witch hunt. This is not how to manage people.
I would ask him what he hopes to achieve by an investigation. He, of course, wants to find out WHO IS GOSSIPING ABOUT HIM. Ask him why. What does he intend to do with that information? Fire the person? Because most companies would not say this is a fireable offense. (Fireable=not a word according to firefox.) Point out that if gossip was truly a problem in his group, he'd know who the instigator was, without any investigation. Those things usually come to light pretty quickly.
Instead, suggest that he let this go. (I know, not likely to happen.) Suggest it again. Congratulate him for being open enough that Jane felt comfortable coming to him directly. Suggest regular staff and 1:1 meetings if he's not having them. These will help people be aware of what is going on and feel more included. This will also allow him to figure out problems and fix them. Tell him that you will be happy to coach him through this process. Because you need to.
Now, this assumes Jane isn't an attention seeking whacko. Because if she is, this is the wrong answer. Heh.